Approaching the thermodynamic limit in equilibrated scale-free networks

TL;DR

This paper investigates how equilibrated scale-free networks approach their theoretical limits as their size increases, revealing significant finite-size effects and different scaling behaviors for various network properties.

Contribution

It provides a detailed analysis of finite-size corrections and scaling behaviors of degree distributions and maximal degrees in equilibrated scale-free networks.

Findings

Finite-size corrections to cutoff scaling are strong even for networks of 10^9 nodes.

Logarithmic correction observed for degenerated graphs with degree distribution ~k^(-3).

Maximal degree distribution converges faster to the limit than the cutoff.

Abstract

We discuss how various models of scale-free complex networks approach their limiting properties when the size N of the network grows. We focus mainly on equilibrated networks and their finite-size degree distributions. Our results show that subleading corrections to the scaling of the position of the cutoff are strong even for networks of order 10^9 nodes. We observe also a logarithmic correction to the scaling for degenerated graphs when the degree distribution follows a power law k^(-3). We study also the distribution of the maximal degree and show that it may have a different scaling than the cutoff and, moreover, it approaches the thermodynamic limit much faster. We present also some results on the cutoff function and the distribution of the maximal degree in equilibrated networks.